Continue that logic, infinite dimensions and the cone would infinitely smaller? Im sure I have that wrong but cool to think about.
Imagine that you have something which depends on many variables (hundreds), and you're trying to predict its behavior based on your previous experience. There is a high chance that the next combination of variable values that you see will be in one of the corners of the many-dimensional cube, because that's where the volume is (the central part of the cube has negligible volume, as we said above). This means that every measurement is in effect an outlier along several dimensions, making predictions very difficult. This is part of the "curse of dimensionality" in statistics. I have seen some people with excellent understanding of mathematics trip themselves up in this area.
But spaces with infinite dimensions are difficult. They are usually required to have a finite norm for all points. Idk how that would affect volume.
If you want to actually have infinite-dimensional volumes, you can't just assign finite values to them in a simple way, or you will have contradictions such as a certain volume being completely covered by a union of things which have 0 volume. In infinite dimensions, you instead have various measures like the Gaussian measure. Feynman's path integrals are a kind of way to assign a value - called amplitude - to an infinite-dimensional manifold (a kind of "volume") of paths. But that takes us well to the side of the idea of the ratio between cube and inscribed figure volumes.